Question

if two vertices of an equilateral triangle are (minus 2 , 0) and 2, 0 find the the third vertices​

Answer 1

As 2 vertices are at equal distance from origun

therefore to would be along y axis

because its along perpendicular bisector passing through origin which is y axis of x axis

As distance between 2 points is (2-(-2) = 2+2 = 4

So let Point on y axis be ( 0,y)

So (0,y) and (2,0) distance between ahold be 4


√ y^2 + 4 = 4

y^2 + 4 = 16

y^2 = 12

y= 2√3

So third vertex is (0,2√3)

Answer 2

Answer: (0, 2√3) and (0, -2√3)

Step-by-step explanation:

The two vertices are (-2, 0) and (2, 0).

The y coordinates are equal to 0. So the side joining these two points goes through the x axis.

Let me find the length of the side of this equilateral triangle, i.e., distance between these points.

$\sqrt{(-2-2)^2+(0-0)^2} \\ \\ \sqrt{(-4)^2+0^2} \\ \\ \sqrt{16+0} \\ \\ \sqrt{16} = 4$

So the sides are 4 units each.

Let me find the midpoint of this side.

$(\frac{-2+2}{2}, \frac{0-0}{2}) \\ \\ (\frac{0}{2}, \frac{0}{2}) \\ \\ (0,0)$

So the midpoint is the origin.

Let me find the altitude of the triangle.

If a side of an equilateral triangle is 'a', then the altitude will be 'a√3 / 2'.

Here, a = 4.

∴ Length of the altitude = 4√3 / 2 = 2√3 units.

Okay. Now see this carefully.

We know that the altitude of an equilateral triangle meets the corresponding side at its midpoint

The side joining (-2, 0) and (2, 0) goes through the x axis. So the altitude will either go through the y axis or be parallel to the y axis.

But as this altitude meets the side at origin, the midpoint, this altitude goes through the y axis.

∴ As it goes through the y axis, the x coordinates of the possible third vertices becomes 0.

We found that the altitude is 2√3 units. So the third vertices are 2√3 units away from the origin (midpoint), perpendicularly from the side joining (-2, 0) and (2, 0).

∴ To get the y coordinate of the third vertex above this side, add 2√3 to the y coordinate of the midpoint, i.e., 0.

And, to get the y coordinate of the third vertex below this side, subtract 2√3 from the y coordinate of the midpoint, i.e., 0.

∴ y coordinates of the possible third vertices above and below the sides are respectively 0 + 2√3 = 2√3 and 0 - 2√3 = -2√3.

∴ Coordinates of the possible third vertices are (0, 2√3) and (0, -2√3).

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Hope this may be helpful.

I began answering this question at 11 am. But when I finished and was going to post it, unfortunately, electricity went off THRICE!!! I'm doing it in PC but my UPS is not working properly. Bad luck! But that's too funny for me!!!

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